This document contains suggested solutions to problems from Tutorial 3 of the mathematics of games. The first problem calculates the expected value of a combined bet on black and a specific group of numbers on a roulette wheel. The solution breaks the problem into probabilities of different outcomes occurring and calculates an expected loss of 14/38 cents per dollar bet. The second problem shows that a mythical slot machine with specific jackpot payouts is a fair game by calculating the expected profit per spin to be zero. The third problem analyzes the expected profit per round of a martingale betting strategy. It derives a formula showing the expected profit is equal to the initial bet multiplied by 1 minus the probability of losing all allowed bets,

1. GEK1544 The Mathematics of Games
Suggested Solutions to Tutorial 3
1. Consider a Las Vegas roulette wheel with a bet of $5 on black (payoff = 1 : 1) and
a bet of $2 on the specific group of 4 (e.g. 13, 14, 16, 17 ; payoff = 8 : 1). What is the
bettor’s expectation on this combined bet?
Suggested solution. Let
A = {outcome is black } and B = {outcome is 13, 14, 16 or 17} .
Then
2
P1 := P (A and B) = P ({outcome is 13 or 17 }) = ,
38
18 − 2
P2 := P (A and [not B]) = P ({outcome is black but not 13 and 17 }) = ,
38
2
P3 := P ([not A] and B) = P ({outcome is 14 or 16 }) = ,
38
P4 := P ([not A] and [not B])
= P ({ outcome is white, 0 , or 00, and not 14 or 16 } )
20 − 2 18
= = .
38 38
These result
X = P1 · (5 + 2 ∗ 8) + P2 · (5 − 2) + P3 · (2 ∗ 8 − 5) + P4 · (−5 − 2)
2 16 2 18
= · (5 + 2 ∗ 8) + · (5 − 2) + · (2 ∗ 8 − 5) + · (−5 − 2)
38 38 38 38
18 20 4 34
= ·5− ·5 + ·2∗8− ·2
38 38 38 38
14
= − .
38
14 1
That is, the bettor’s expected loss is · ≈ 5.263 cents for every dollar bet.
38 5 + 2
2. A mythical slot machine has three wheels, each containing ten symbols. On each
wheel there is 1 JACKPOT symbol and 9 other non-paying symbols. You put in 1 silver
dollar ($1) in the slot and the payoffs are as follows :
3 JACKPOT symbols – $487 in silver is returned (including your $1).
2 JACKPOT symbols – $10 in silver is returned (including your $1).
1 JACKPOT symbols – $1 in silver is returned (you get your wager back!).
Define what it would mean to say that this slot machine is fair and then show that it is
indeed a fair “one-armed bandit” !

2. Suggested solution. Assuming that the slot machine is random.
1 9 9 9 1 9 9 9 1 243
P1 = P (1 Jackpot symbol exactly) = · · + · · + · · = ;
10 10 10 10 10 10 10 10 10 1000
1 1 9 1 9 1 9 1 1 27
P2 = P (2 Jackpot symbols exactly) = · · + · · + · · = ;
10 10 10 10 10 10 10 10 10 1000
1 1 1 1
P3 = P (3 Jackpot symbols) = · · = ;
10 10 10 1000
9 9 9 729
pN = P (No Jackpot symbol at all) = · · = = 1 − (P1 + P2 + P3 ) [please verify] .
10 10 10 1000
The profits are 0, 9 and 486, respectively. We infer that
27 ∗ 9 + 486 − 729
X = P1 ∗ 0 + P2 ∗ 9 + P3 ∗ 486 + pN ∗ (−1) = = 0.
1000
3. The martingale strategy had the gambler double his bet after every loss, so that
the first win would recover all previous losses plus win a profit equal to the original stake.
Since a gambler with infinite wealth will with probability 1 eventually flip heads, the
Martingale betting strategy was seen as a sure thing by those who practised it.
Of course, none of these practitioners in fact possessed infinite wealth, and the
exponential growth of the bets would eventually bankrupt those who choose to use the
Martingale. Moreover, it has become impossible to implement in modern casinos, due to
the betting limit at the tables. Because the betting limits reduce the casino’s short term
variance, the Martingale system itself does not pose a threat to the casino, and many will
encourage its use, knowing that they have the house advantage no matter when or how
much is wagered.
Let one ‘round’ be defined as a sequence of consecutive losses followed by a win,
or consecutive losses resulting in bankruptcy of the gambler.
After a win, the gambler “resets” and is considered to have started a new round.
A continuous sequence of martingale bets can thus be partitioned into a sequence of
independent rounds. We will analyze the expected value of one round.
Let be the probability of losing (e.g. betting for “even” for roulette has 20/38
chance of losing). Let y be the amount of the commencing bet, and n the finite number
of bets you can afford to lose. Show that the expected profit (a loss if the number is
negative) per round is given by
y · [1 − (2 )n ] .
Suggested solution. In the sequence of numbers
20 y = 1 ∗ y, 2y, 22 y, · · ·, 2n−1 y, 2n y,
n
the chance that the better loses the first n games is . The loss is
2n − 1
1 ∗ y + 2y + 22 y + · · · + 2n−1 y = y 1 + 21 + · · · + 2n−1 = y · = y (2n − 1) .
2−1
The chance that the better does not lose all n games is (1 − n ). In this case, disregard
where is the win in the ‘round’, the profit is y. Hence
n
X= [−(y (2n − 1))] + (1 − n
) y = y · [1 − (2 )n ] .
1
As a side note, X < 0 once the probability of losing > 2.